Question

Solve the initial-boundaryvalue problem. In some of these exercises, Theorem 11.3.5(b) or Exercise 11.3.35 will simplify the computation of the coefficients in the Fourier sine series. $$ \begin{array}{l} \text u_{t t}=9 u_{x x}, \quad 00 \\ u(0, t)=0, \quad u(1, t)=0, \quad t>0, \\ u(x, 0)=0, \quad u_{t}(x, 0)=x(1-x), \quad 0 \leq x \leq 1 \end{array} $$

Short answer

Question: Find the function u(x, t) that specifies the solution to the given initial-boundary value problem involving a partial differential equation: PDE: $$ u_{xx} = 9u_{tt}$$ Boundary Conditions: $$ u(0,t) = u(1,t) = 0 $$ Initial Conditions: $$ u(x,0) = 0 $$ and $$ u_t(x,0) = x(1-x) $$ Answer: The solution to this problem is given by: $$ u(x, t) = \sum_{n=1}^\infty \sin(n\pi x)B_n\sin(n\pi t) $$ Where $$ B_n = \frac{2}{n\pi}\int_0^1 x(1-x)\sin(n\pi x)\,dx $$

Step-by-step solution

Apply separation of variables

Assume that the solution u(x,t) can be separated into two functions U(x) and T(t) such that u(x,t) = U(x)T(t). We plug this into the given PDE: $$ U''(x)T(t) = 9U(x)T''(t) $$ Now, divide both sides by 9U(x)T(t): $$ \frac{U''(x)}{9U(x)} = \frac{T''(t)}{T(t)} $$ Since the left side only depends on x and the right side only depends on t, both sides must be equal to a constant, say: $$ \frac{U''(x)}{9U(x)} = \frac{T''(t)}{T(t)} = -\lambda $$

Solve the eigenvalue problems

Now, we have two ordinary differential equations: $$ U''(x) + 9\lambda U(x) = 0 $$ and $$ T''(t) + \lambda T(t) = 0 $$ The boundary conditions are: $$ U(0) = U(1) = 0 $$ From these boundary conditions and the ODE for U(x), we can determine that the eigenvalues are: $$ \lambda_n = (n\pi)^2, \quad n = 1,2,3,... $$ with corresponding eigenfunctions: $$ U_n(x) = \sin(n\pi x), \quad n = 1,2,3,... $$

Solve for the time-dependent coefficient T(t)

Using the eigenvalues found in Step 2, we can solve the ODE for T(t): $$ T''(t) + (n\pi)^2 T(t) = 0 $$ The general solution of this ODE is given by: $$ T_n(t) = A_n\cos(n\pi t) + B_n\sin(n\pi t), \quad n = 1,2,3,... $$ Now, we can write the general solution $$ u(x, t) = \sum_{n=1}^\infty \sin(n\pi x)[A_n\cos(n\pi t) + B_n\sin(n\pi t)] $$

Apply initial conditions to find the coefficients

We need to find the coefficients A_n and B_n using the initial conditions: $$ u(x, 0) = \sum_{n=1}^\infty A_n\sin(n\pi x) = 0 $$ The initial condition u(x,0) = 0 tells us that A_n = 0. The initial condition for the derivative u_t(x,0) is: $$ u_{t}(x, 0) = \sum_{n=1}^\infty B_n n\pi\cos(n\pi x) = x(1-x) $$ We now apply Theorem 11.3.5(b) to find B_n: $$ B_n = \frac{2}{n\pi}\int_0^1 x(1-x)\sin(n\pi x)\,dx $$

Final solution

Now, using the values of A_n and B_n, we can write the final solution: $$ u(x, t) = \sum_{n=1}^\infty \sin(n\pi x)B_n\sin(n\pi t) $$ Where $$ B_n = \frac{2}{n\pi}\int_0^1 x(1-x)\sin(n\pi x)\,dx $$

Key concepts

Fourier Sine Series

In the realm of mathematics and specifically, in the study of initial-boundary value problems, the Fourier sine series is an indispensable tool used to express a function defined on a finite interval as a sum of sine functions. It's a representation that allows a function to be decomposed into a series of simpler, sinusoidal components, each of which oscillates with a different frequency. This decomposition is particularly useful for solving partial differential equations (PDEs) with certain boundary conditions.

The Fourier sine series takes the form:
\[ u(x) = \frac{a_0}{2} + sum_{n=1}^{fty} a_n sin(n pi x)\]
where the coefficients \( a_n \) are found using the orthogonality of sine functions over the interval and the specific initial or boundary conditions of the problem at hand. In the exercise you're studying, calculating the Fourier coefficients involves integrating a function against the sine functions that form the basis of the Fourier series. This step is key in linking the initial conditions of the problem to the spatial sine functions that will construct the solution.

Separation of Variables

The method of separation of variables is a powerful technique used for solving partial differential equations. This method is applicable when a PDE and its boundary conditions possess a certain symmetry that allows the variables to be separated. In essence, the complex problem involving multiple variables is broken down into simpler, ordinary differential equations (ODEs) that are easier to solve.

The process involves assuming that the solution to the PDE can be written as a product of functions, each depending on only one of the independent variables. For the problem under consideration, this means writing the solution \(u(x, t)\) as a product of two separate functions: \(U(x)\) and \(T(t)\). By doing this, the PDE initially involving both x and t can be segregated into two ODEs on x and t, respectively. Once this separation has been achieved, each ODE can be solved individually, often by using techniques suitable for ODEs, such as finding the eigenvalues and eigenfunctions.

Eigenvalue Problems

An eigenvalue problem is a type of problem that appears frequently when solving linear differential equations and PDEs, especially when employing the separation of variables method. In the context of solving a PDE, the spatial part of the separation leads to an ordinary differential equation with boundary conditions that form an eigenvalue problem.

An eigenvalue problem has the general form \[L[u(x)] = lambda u(x)\], where \(L\) is a differential operator, \(u(x)\) is the eigenfunction, and \(lambda\) is the eigenvalue. The boundaries of the domain play a crucial role, as they determine whether a certain value of \(lambda\) and the corresponding function \(u(x)\) are permissible, i.e., satisfy both the differential equation and the boundary conditions. Solving this problem yields a discrete spectrum of eigenvalues and associated eigenfunctions, which are utilized in expanding the solution to the original PDE in terms of sine and cosine functions.

Partial Differential Equations

Partial differential equations (PDEs) form a cornerstone of the mathematical modeling of processes in science and engineering. These equations involve functions of several variables and their partial derivatives. PDEs can describe a wide array of phenomena such as heat transfer, wave motion, fluid dynamics, and quantum mechanics.

Unlike ordinary differential equations, which involve derivatives with respect to only one variable, PDEs contain derivatives with respect to multiple independent variables. The example given in the exercise involves a second-order PDE with time (t) and one spatial dimension (x). Solving a PDE usually requires conditions to be specified on the boundaries of the domain, known as boundary conditions, as well as initial conditions. In the provided example, the initial-boundary value problem is solved by employing various mathematical techniques such as separation of variables and the Fourier sine series method to conform to the prescribed conditions and obtain a detailed description of the evolution of the system over time.